In the rapidly expanding field of wireless digital communications, demand continues to increase for reliable wireless systems that have a high spectral efficiency. Accordingly, the use of Orthogonal Frequency Division Multiplexing (OFDM) technology is increasing within wireless applications such as cellular and Personal Communication Systems (PCS). OFDM has a high tolerance to multipath signals and is spectrally efficient which makes it a good choice for future wireless communication systems. Utilization of OFDM technology will continue to be significant as such technology is incorporated within new standards currently being defined.
More specifically, OFDM is a special form of multicarrier modulation that uses Digital Signal Processor (DSP) algorithms like Inverse Fast Fourier Transform (IFFT) to generate waveforms that are mutually orthogonal and Fast Fourier Transform (FFT) for demodulation operations. In a typical OFDM signal format, the IFFT modulator operation may be represented by Equation 1 (Eq.1), and the FFT demodulator operation may be represented by Equation 2 (Eq.2).
                              x          n                =                              ∑                                          k                =                                                      -                                          (                                              N                        -                        1                                            )                                                        /                  2                                                            k                ≠                0                                                                    (                                  N                  -                  1                                )                            /              2                                ⁢                                    X              k                        ·                          ⅇ                              j                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                                  n                  /                  N                                                                                        Eq        .                                  ⁢        1                                          X          k                =                              1            N                    ⁢                                    ∑                              n                =                                                      -                                          (                                              N                        -                        1                                            )                                                        /                  2                                                                              (                                      N                    -                    1                                    )                                /                2                                      ⁢                                          x                n                            ·                              ⅇ                                                      -                    j                                    ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                      kn                    /                    N                                                                                                          Eq        .                                  ⁢        2            
Equations 1 and 2 can be written in vector format as x=QX and X=Fx, where Q and F is the IFFT and FFT matrixes with elements qn,k and fn,k, respectively, and in accordance with Equations 3 (Eq.3) and 4 (Eq.4).qn,k=ej2πkn/N  Eq. 3
                              f                      n            ,            k                          =                              1            N                    ⁢                      ⅇ                                          -                j                            ⁢                                                          ⁢              2              ⁢                                                          ⁢              π              ⁢                                                          ⁢                              kn                /                N                                                                        Eq        .                                  ⁢        4            
While OFDM has existed for some time, OFDM has only recently become widely implemented in high-speed digital communications due to advancements in Very Large-Scale Integrated Circuit (VLSI) technologies and related DSP design and fabrication. OFDM has gained a lot of interest in diverse digital communication applications due to its favorable properties like high spectral efficiency, robustness to channel fading, immunity to impulse interference, uniform average spectral density, and capability of handling very strong echoes.
OFDM technology is now used in many new broadband communication schemes and many other wireless communication systems. However, there are some concerns with regard to OFDM. Such concerns include high Peak-to-Average Power Ratio (PAPR) and frequency offset. High PAPR causes saturation in power amplifiers, leading to intermodulation products among the subcarriers and disturbances of out-of-band energy. Therefore, it is desirable to reduce the PAPR. In order to meet the out-of-band emissions requirements, a power amplifier and other components with this high PAPR input are required to provide good linearity in a large dynamic range. This makes the power amplifier one of the most expensive components within the communication system. The high PAPR also means that the power amplifier operation has low power efficiency that reduces battery life for related mobile stations.
Another concern of OFDM is that the peak of the signal can be up to N times the average power (where N is the number of carriers). These large peaks increase the amount of intermodulation distortion resulting in an increase in the error rate. The average signal power must be kept low in order to prevent the transmitter amplifier limiting. Minimizing the PAPR allows a higher average power to be transmitted for a fixed peak power, improving the overall signal to noise ratio at the receiver. It is therefore important to reduce or otherwise minimize the PAPR. The plain PAPR without any reduction for a given OFDM symbol can be defined by Equation 5 (Eq.5). Whereas, the obtained PAPR with a reduction algorithm fPAPR for a given OFDM symbol can be defined by Equation 6 (Eq.6).
                              PAPR          p                =                                                                            x                                            ∞              2                                                      E                ⁡                                  [                                                                                  x                                                              2                    2                                    ]                                            /              N                                =                                                    max                n                            ⁢                              {                                                                                                x                      n                                                                            2                                }                                                                    E                ⁡                                  [                                                                                  x                                                              2                    2                                    ]                                            /              N                                                          Eq        .                                  ⁢        5                                          PAPR          ⁡                      (                          f              PAPR                        )                          =                                                                                                                  f                    PAPR                                    ⁡                                      (                    x                    )                                                                              ∞              2                                                      E                ⁡                                  [                                                                                                                                    f                          PAPR                                                ⁡                                                  (                          x                          )                                                                                                            2                    2                                    ]                                            /              N                                =                                                    max                n                            ⁢                              {                                                                                                                        f                        PAPR                                            ⁡                                              (                                                  x                          n                                                )                                                                                                  2                                }                                                                    E                ⁡                                  [                                                                                                                                    f                          PAPR                                                ⁡                                                  (                          x                          )                                                                                                            2                    2                                    ]                                            /              N                                                          Eq        .                                  ⁢        6            
The PAPR of an OFDM signal can be reduced by several different PAPR reduction methods. Such methods can be classified into two groups including Constellation Shaping (CS) (e.g., distortionless or active constellation expansion) and Tone Reservation (TR). With CS methods, the modulation constellation is changed such that the obtained PAPR is less than the required value with the satisfied channel error criteria. With TR methods, the reserved tones are assigned with such values that the obtained PAPR is less than the required value with the satisfied channel error criteria. In the tone reservation method, the basic idea is to reserve a small set of tones for PAPR reduction. Computing the values for these reserved tones that minimize the PAPR can advantageously be formulated as a convex problem and can be solved exactly. The amount of PAPR reduction depends on the number of reserved tones, their locations within the frequency vector, and the amount of complexity. Known TR methods typically have high complexity and involve iterations that convert the signal back and forth between the frequency domain and the time domain. Other methods of reducing PAPR are also possible but they affect signal quality or Error-Vector Magnitude (EVM).
It is, therefore, desirable to provide an optimal TR method that can reduce the PAPR of OFDM signals input to power amplifiers. Such an optimal TR method should reduce the peaks of the compounded input signals such that a less expensive power amplifier can be utilized with out-of-band emissions still being fully controlled. This optimal TR method should also be relatively inexpensive and any degradation in terms of in-band signal quality should be within an acceptable range.